= = 6, and let another be described similar to this, upon a right line to its greatest side; if it be constructed so that the <s adjacent to this right line shall be to those which are adjacent to the side to this, the polygons shall be =; but if it be so constructed, that they are the <s which are adjacent to the sides denoted by 4, 2, or 1 the polygons shall be to one another, as 36: 16, or as 36 4, or as 36: 1. PROP. XXI. THEOR. to Rectilinear figures, which are similar to the same figure, are similar to one another. Since each of the given figures is similar to the same figure, they are equiangular with it, and have the sides about the angles proportional, the given figures are equiangular with one another, and have the sides about those angles proportional; .. they are similar to one another. PROP. XXII. THEOR. If four right lines be proportional, the similar rectilinear figures similarly described on them are also proportional. And if similar rectilinear figures similarly described on four right lines be proportional, the right lines themselves will be proportional. : PART 1. Let a third proportional be taken to the first and second, as also to the third and fourth. Then ex æquali the first its third propor. the third its third proportional; but the figures on the first and second are to one another as the first to its third propor. (Cor. 1. 20. 6) and the figures on the third and fourth are to one another as the third to its third propor., .. the figures on the first and second are to one another as the figures on the third and fourth, i. e. the figures, &c. are proportional. : PART 2. The same construction being made, since the figures are propor. to one another, the first its third propor, the third its third propor... the first: the second the third: the fourth, i. e. the lines are proportional, PROP. XXIII. THEOR. Equiangular parallelograms are to one another in a ratio compounded of the ratios of their sides. Let the given parallelograms be so placed that equal s may be vertically opposite, and complete the parallelogram of which those sides of the given parallelograms, not in directum, are sides. Then the first par. is to the constructed one as a side of the first to a side of the second in directum with it, and the constructed par. is to the second as another side of the first to another side of the second in directum with it, .. the first par. : the second in a ratio compounded of the ratios of those sides. Cor. 1. Two right lines can be found which are to one another in the ratio of the given parallelograms. Those two right lines will be the base of the first, and a fourth proportional to the altitude of the first, the altitude of the second, and the base of the second. For the base of the first will be to this fourth proportional in a ratio compounded of the ratios of the base of the first to the base of the second, and of the base of the second to this fourth proportional, (Def. 12. 5.) but this latter is equal to the ratio of the altitudes; the base of the first is to this fourth proportional in a ratio compounded of the ratios of the sides. Cor. 2. Triangles, which have one angle in each equal, are to one another in a ratio compounded of the ratios of the sides containing those equal angles. Cor. 3. Any parallelograms or triangles are to one another in a ratio compounded of the ratios of their bases and altitudes; for they are to the rectangles or to the right angled triangles on equal bases, and of the same altitudes. PROP. XXIV. THEOR. In any parallelogram the parallelograms which ore about the diagonal are similar to the whole and to one another. Since each of the parallelograms about the diagonal has an common with the whole parallelogram, they are equiangular (Cor. 2. 34. 1.) with it. And since either of the As into which the diagonal divides the whole parallelogram is similar to one of the As into which it divides those about the diagonal, the sides about those equal As are proportional; .. &c. PROP. XXV. PROB. To construct a rectilinear figure. equal to a given one, and similar to another. = = To any side of the figure to which a similar one is to be constructed, apply a rectangle to it (Cor. 1. 45. 1), and to either side of this rectangle, adjacent to the side to which it is applied, apply another rectangle to the other given figure; and find a mean proportional between the two sides of those constructed rectangles, that are in directum; the figure constructed on this mean proportional similar to the one required, and similarly posited, will be equal to the other given one. Those two similar figures are to each other in the duplicate ratio, their homologous sides, i. e. by constr. in the ratio of the sides of the rectangles, to which a mean proportional was found, i. e. as the rectangles, i. e. as the two given figures, which (by contr.) are to the rectangles. Therefore the given figure, to which it was required to construct one similar, bears the same ratio to the other given figure and to the constructed figure. PROP. XXVI. THEOR. If similar and similarly posited parallelograms have a common angle, they are about the same diagonal. If possible let them not be about the same diagonal; then the diagonal of the greater parallelogram will cut a side of the less, not at the vertex of the angle which is opposite to the common angle, but in some other point; from this point, in which it cuts one of the sides of the less par., draw a right line parallel to the other side. Then the parallelogram thus formed (i. e. part of the less) will be similar to the greater (24. 6) and .... to the less,.. one side of the less bears the same ratio to two unequal right lines, which is absurd. Therefore, &c. PROPS. XXVII. XXVIII. XXIX. omitted. PROP. XXX. PROB. To cut a given finite right line in extreme and mean ratio. By extreme and mean ratio is meant, so that the whole line shall be to the greater segment as the greater segment is to the less (Def. 4. B. 6)... the rectangle under the whole line and lesser segment shall be to the 2 of the greater, for the greater segment is a mean proportional between the given line and lesser; if .. you cut the given line, as in Prop. XI. Book 2. the thing is done. Note 1. If this series be continued in infinitum, it is evident from Prop. 11. of this Book, that the whole line will be a mean proportional between the smaller segment and the sum of the series. And the successive terms of of the series will be found by taking each consequent from its antecedent. Cor. Prop. 11. B. 2. Note 2. It will also be seen, by the doctrine of ratios, that the successive terms of the series will be found by taking each consequent from its antecedent; for in any analogy the first term is to the second as the difference between the first and third is to the difference between the second and fourth. But when a line is cut in extreme and mean ratio, the greater segment: the lesser : : the whole line the greater segment; the greater segment the lesser the difference between the whole and greater, i. e. as the lesser the difference between the greater and lesser; .. the lesser segment is a mean proportional between the greater and the difference between the segments. Note 3. From those principles, we may describe on a given right line a right angled A in which the 2 of one side about the right shall be = to the rectangle under the hypothenuse and the other side, and the three sides, .. shall be in continued proportion. Note 4. It also follows that if one side of a A be divided in extreme and mean ratio, and from the point of section a right line be drawn to the opposite, and another line parallel to the side conterminous with the lesser segment, the two As thus cut off shall be, and shall have the sides about theirs reciprocally proportional. PROP. XXXI. THEOR. If on the sides of a right angled triangle any similar rectilineal figures be similarly described, the figure described on the side subtending the right angle is equal to the sum of the figures on the other two sides which contain the right angle. : From the right draw a perpendicular to the hypothenuse; then the hypothenuse: one of the sides about the right that side the adjacent segment of the hypothenuse (Cor. 8. 6), .. the figure upon the hypothenuse to the similar figure upon that side about the right the hypoth. the segment adjacent to that side, and therefore the figure upon the hypothenuse the difference between itself and the figure upon that side the hypoth. : `the difference between itself and the segment adjacent to that side, i. e. :: the hypoth, the segment adjacent to the other side about the right; but the figure upon the hypoth. : the similar figure upon this other side about the right the hypoth this difference, i. e. : it is to the segment adjacent to this other side,.. the figure upon this other side about the right < is to the difference between the similar figures upon the hypothenuse and the first mentioned side about the right <. Therefore, &c. : Cor. Given any number of similar rectilineal figures, a similar figure can be found equal to their sum, by Cor. 2. Prop. 47. B. 1. and by Cor. 3. Prop. 47. B. 1. a figure can be found equal to the difference between two given similar figures. |